@article{CML_2011__3_3_445_0, author = {Lee Keyfitz, Barbara}, title = {Singular shocks: retrospective and prospective}, journal = {Confluentes Mathematici}, pages = {445--470}, publisher = {World Scientific Publishing Co Pte Ltd}, volume = {3}, number = {3}, year = {2011}, doi = {10.1142/S1793744211000424}, language = {en}, url = {http://www.numdam.org/articles/10.1142/S1793744211000424/} }
TY - JOUR AU - Lee Keyfitz, Barbara TI - Singular shocks: retrospective and prospective JO - Confluentes Mathematici PY - 2011 SP - 445 EP - 470 VL - 3 IS - 3 PB - World Scientific Publishing Co Pte Ltd UR - http://www.numdam.org/articles/10.1142/S1793744211000424/ DO - 10.1142/S1793744211000424 LA - en ID - CML_2011__3_3_445_0 ER -
%0 Journal Article %A Lee Keyfitz, Barbara %T Singular shocks: retrospective and prospective %J Confluentes Mathematici %D 2011 %P 445-470 %V 3 %N 3 %I World Scientific Publishing Co Pte Ltd %U http://www.numdam.org/articles/10.1142/S1793744211000424/ %R 10.1142/S1793744211000424 %G en %F CML_2011__3_3_445_0
Lee Keyfitz, Barbara. Singular shocks: retrospective and prospective. Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 445-470. doi : 10.1142/S1793744211000424. http://www.numdam.org/articles/10.1142/S1793744211000424/
[1] M. B. Allen III, G. A. Behie and J. A. Trangenstein, Multiphase Flow in Porous Media: Mechanics, Mathematics and Numerics (Springer-Verlag, 1988).
[2] J. B. Bell, J. A. Trangenstein and G. R. Shubin, Conservation laws of mixed type describing three-phase flow in porous media, SIAM J. Appl. Math. 46 (1986) 1000–1023.
[3] J. H. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Quart. Appl. Math. XVIII (1960) 191–204.
[4] V. A. Borovikov, On the decomposition of a discontinuity for a system of two quasi- linear equations, Trans. Moscow Math. Soc. 27 (1972) 53–94.
[5] F. Bouchut, On zero pressure gas dynamics, in Advances in Kinetic Theory and Com- puting, Selected Papers, Advances in Mathematics for Applied Sciences, Vol. 22 (World Scientific, 1994), pp. 171–190.
[6] A. Bressan, Hyperbolic Systems of Conservation Laws: The One-Dimensional Cauchy Problem (Oxford Univ. Press, 2000).
[7] G. F. Carrier, On the nonlinear vibration problem of the elastic string, Quart. Appl. Math. 3 (1945) 157–165.
[8] Y. Chen, J. Glimm, D. H. Sharp and Q. Zhang, A two phase flow model of the Rayleigh–Taylor mixing zone, LASL preprint, LA-UR 95-3526.
[9] J.-F. Colombeau, Multiplication of distributions, Bull. Amer. Math. Soc. 23 (1990) 251–268.
[10] C. M. Dafermos and R. J. DiPerna, The Riemann problem for certain classes of hyperbolic systems of conservation laws, J. Differential Equations 20 (1976) 90–114.
[11] R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985) 223–270.
[12] H. Freistühler, Rotational degeneracy of hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 113 (1991) 39–64.
[13] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965) 95–105.
[14] E. L. Isaacson, D. Marchesin, B. Plohr and J. B. Temple, The Riemann problem near a hyperbolic singularity, I, SIAM J. Appl. Math. 48 (1988) 1009–1032.
[15] J. J. Cauret, J. F. Colombeau and A. Y. Le Roux, Discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations, J. Math. Anal. Appl. 139 (1989) 552–573.
[16] B. L. Keyfitz, Change of type in simple models of two-phase flow, in Viscous Pro- files and Numerical Approximation of Shock Waves, ed. M. Shearer (SIAM, 1991), pp. 84–104.
[17] B. L. Keyfitz, Multiphase saturation equations, change of type and inaccessible regions, in Proc. Oberwolfach Conf. on Porous Media, eds. J. Douglas, C. J. van Duijn and U. Hornung (Birkhäuser, 1993), pp. 103–116.
[18] B. L. Keyfitz, Conservation laws, delta shocks and singular shocks, in Nonlinear Theory of Generalized Functions, eds. M. Grosser, G. Hörmann, M. Kunzinger and M. Oberguggenberger (Chapman & Hall/CRC Press, 1999), pp. 99–111.
[19] B. L. Keyfitz, Mathematical properties of nonhyperbolic models for incompressible two-phase flow, in Proc. Fourth Int. Conf. Multiphase Flow, New Orleans (CD ROM), ed. E. E. Michaelides, ICMF 2001, Tulane University, 2001.
[20] B. L. Keyfitz and H. C. Kranzer, A system of hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 72 (1980) 219–241.
[21] B. L. Keyfitz and H. C. Kranzer, The Riemann problem for a class of hyperbolic con- servation laws exhibiting a parabolic degeneracy, J. Differential Equations 47 (1983) 35–65.
[22] B. L. Keyfitz and H. C. Kranzer, A viscous approximation to a system of conser- vation laws with no classical Riemann solution, in Nonlinear Hyperbolic Problems, eds. C. Carasso et al., Lecture Notes in Mathematics, Vol. 1402 (Springer, 1989), pp. 185–197.
[23] B. L. Keyfitz and H. C. Kranzer, Spaces of weighted measures for conservation laws with singular shock solutions, J. Differential Equations 118 (1995) 420–451.
[24] B. L. Keyfitz, R. Sanders and M. Sever, Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow, Disc. Cont. Dynam. Syst. 3 (2003) 541–563.
[25] B. L. Keyfitz, M. Sever and F. Zhang, Viscous singular shock structure for a nonhy- perbolic two-fluid model, Nonlinearity 17 (2004) 1731–1747.
[26] B. L. Keyfitz and C. Tsikkou, Conserving the wrong variables in gas dynamics: A Riemann solution with singular shocks, Quart. Appl. Math. in preparation.
[27] H. C. Kranzer and B. L. Keyfitz, A strictly hyperbolic system of conservation laws admitting singular shocks, in Nonlinear Evolution Equations that Change Type, eds. B. L. Keyfitz and M. Shearer, IMA, Vol. 27 (Springer, 1990), pp. 107–125.
[28] S. N. Kruˇzkov, First-order quasilinear equations in several independent variables, Math. USSR — Sbor. 10 (1970) 217–243.
[29] H. A. Levine and B. D. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math. 57 (1997) 683–730.
[30] M. Mazzotti, Local equilibrium theory for the binary chromatography of species subject to a generalized Langmuir isotherm, Indust. Eng. Chem. Res. 45 (2006) 5332–5350.
[31] M. Mazzotti, Non-classical composition fronts in nonlinear chromatography — Delta- shock, Indust. Eng. Chem. Res. 48 (2009) 7733–7752.
[32] M. Mazzotti, A. Tarafder, J. Cornel, F. Gritti and G. Guiochon, Experimental evi- dence of a delta-shock in nonlinear chromatography, J. Chromatography A 1217 (2010) 2002–2012.
[33] M. Nedeljkov, Shadow waves: Entropies and interactions for delta and singular shocks, Arch. Rational Mech. Anal. 197 (2010) 489–537.
[34] H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC’s of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (1997) 1044–1081.
[35] H.-K. Rhee, R. Aris and N. R. Amundson, First-Order Partial Differential Equations: Vol. II, Theory and Application of Hyperbolic Systems and Quasilinear Equations (Prentice-Hall, 1989).
[36] P. Rosenau, Evolution and breaking of ion-acoustic waves, Phys. Fluids 31 (1988) 1317–1319.
[37] D. G. Schaeffer, S. Schecter and M. Shearer, Nonstrictly hyperbolic conservation laws with a parabolic line, J. Differential Equations 103 (1993) 94–126.
[38] D. G. Schaeffer and M. Shearer, The classification of 2 × 2 systems of non-strictly hyperbolic conservation laws, with application to oil recovery, Comm. Pure Appl. Math. XL (1987) 141–178.
[39] S. Schecter, Existence of Dafermos profiles for singular shocks, J. Differential Equa- tions 205 (2004) 185–210.
[40] M. Sever, Viscous structure of singular shocks, Nonlinearity 15 (2002) 705–725.
[41] M. Sever, Distribution solutions of nonlinear systems of conservation laws, Mem. Amer. Math. Soc. 190 (2007) 1–163.
[42] M. Sever, The Cauchy problem for a model problem with singular shocks, J. Hyperbolic Differential Equations 7 (2010) 1–66.
[43] J. A. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, 1983).
[44] H. B. Stewart and B. Wendroff, Two-phase flow: Models and methods, J. Comput. Phys. 56 (1984) 363–409.
[45] D. Tan, T. Zhang and Y.-X. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112 (1994) 1–32.
[46] V. Vinod, Structural stability of Riemann solutions for a multiphase kinematic con- servation law model that changes type, PhD thesis, University of Houston, Houston, Texas, 77204-3476, 1992.
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